Special group: Difference between revisions
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==Definition== | ==Definition== | ||
A [[group of prime power order]] is termed '''special''' if its [[center]], [[ | A [[group of prime power order]] is termed '''special''' if its [[center]], [[derived subgroup]] and [[Frattini subgroup]] all coincide. It turns out that in this case, the center must be an [[elementary abelian group]]. | ||
Sometimes, the term '''special''' also includes the case of [[elementary abelian group]]s. Under this definition, a group is special if it is either special in the above sense ''or'' it is elementary abelian. | |||
{{further|[[Special implies center is elementary abelian]]}} | |||
==Relation with other properties== | ==Relation with other properties== | ||
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===Weaker properties=== | ===Weaker properties=== | ||
* [[Group of | * [[Group of nilpotency class two]] | ||
* [[UL-equivalent group]] | |||
Latest revision as of 01:58, 12 June 2011
The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
View other prime-parametrized group properties | View other group properties
Definition
A group of prime power order is termed special if its center, derived subgroup and Frattini subgroup all coincide. It turns out that in this case, the center must be an elementary abelian group.
Sometimes, the term special also includes the case of elementary abelian groups. Under this definition, a group is special if it is either special in the above sense or it is elementary abelian.
Further information: Special implies center is elementary abelian